Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra

Movshev, M. V.; Schwarz, Albert; Xu, Renjun

doi:10.1016/j.nuclphysb.2011.08.023

Cited by 24 publications

(56 citation statements)

References 6 publications

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“…At the first stage, the invariant higher central extensions of super Minkowski space-time are classified by the invariant higher Lie algebra cocycles of the corresponding ordinary translational supersymmetry super Lie algebras. These have been classified by a variety of methods [22][23][24][25][26][27][28] and constitute the "old brane scan" (Table 1). In particular, in the "critical" dimensions one finds:…”

Section: The Brane Bouquetmentioning

confidence: 99%

“…These Green–Schwarz‐type sigma models for super p ‐branes turn out to have a deep supergeometric origin which implies the remakable fact that they are mathematically classified by the Spin‐invariant super Lie algebra cohomology of super Minkowski space‐times . This classification has come to be known as the old brane scan , reproduced in Table above.…”

Section: Introductionmentioning

confidence: 99%

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The Rational Higher Structure of M‐theory

Fiorenza

Sati

Schreiber

2019

Fortschritte der Physik

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We review how core structures of string/M‐theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion‐subgroups in brane charges and focuses on tangent spaces of super space‐time. Already at this level, super homotopy theory discovers all super p‐brane species, their intersection laws, their M/IIA‐, T‐ and S‐duality relations, their black brane avatars at ADE‐singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D‐branes it recovers twisted K‐theory, rationally, but for the M‐branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M‐theory in super homotopy theory.

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Section: The Brane Bouquetmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Rational Higher Structure of M‐theory

Fiorenza

Sati

Schreiber

2019

Fortschritte der Physik

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“…The infinitesimal translation symmetries of this object form a Lie superalgebra, which we call the 'supertranslation algebra', T . The cohomology of this Lie superalgebra is interesting and apparently rather subtle [14,41,42]. We shall see that its 3rd cohomology is nontrivial in dimensions n + 2 = 3, 4, 6 and 10, thanks to the 3-ψ's rule.…”

Section: The Supertranslation Lie 2-superalgebrasmentioning

confidence: 82%

“…Techniques to compute it have been described by Brandt [14], who applied them in dimension 5 and below. Movshev, Schwarz and Xu [41,42] showed how to augment these techniques using computer algebra systems, such as LiE [19], and fully describe the cohomology in dimensions less than 11 in this way.…”

Section: The Supertranslation Lie 2-superalgebrasmentioning

confidence: 99%

Division algebras and supersymmetry III

Huerta

2012

Advances in Theoretical and Mathematical Physics

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Recent work applying higher gauge theory to the superstring has indicated the presence of 'higher symmetry'. Infinitesimally, this is realized by a 'Lie 2-superalgebra' extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a 'Lie 2-supergroup' extending the Poincaré supergroup in the same dimensions.Briefly, a 'Lie 2-superalgebra' is a two-term chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above. Because this 3-cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2-supergroup integrating the Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincaré supergroup.

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“…Applying Schur's lemma and the information about the Hilbert series we can find the action of the differential on the irreducible components of A N,q ; this allows us to find the so(10)-action on H N,q for small N, q. (We use a version of the "maximal propagation principle" [5], [6] in this consideration.) These results allow us to guess the representatives of the cohomology classes of generators.…”

mentioning

confidence: 99%

Cohomology Ring of the BRST Operator Associated to the Sum of Two Pure Spinors

Mikhailov

Schwarz

2013

Mod. Phys. Lett. A

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In the study of the Type II superstring, it is useful to consider the BRST complex associated to the sum of two pure spinors. The cohomology of this complex is an infinite-dimensional vector space. It is also a finite-dimensional algebra over the algebra of functions of a single pure spinor. In this paper we study the multiplicative structure.The central object of the pure spinor formalism is the BRST operator Q, which involves nonlinearly constrained ghosts. For example, the Type IIB superstring uses two ghosts λ L and λ R , which are spinors of so(10) satisfying the pure spinor constraint:Roughly speaking, the BRST structures are in one-to-one correspondence with SUGRA backgrounds. For a given background, the cohomology of Q describes its infinitesimal deformations. It was shown in [1,2] that the knowledge of the cohomology of the simplified differentialwhere θ is an odd spinor variable is useful for understanding the cohomology of Q. The cohomology groups of Q (0) were calculated in [2] as modules over the ring of polynomials of λ α L , λ α R . In the present paper we will study the multiplication of the cohomology classes, and calculate the cohomology as a ring. This allows us to simplify the description of cohomology. We can use the multiplication in cohomology to obtain information about cohomology of more complicated differentials.Let us consider the ring A = C[λ L , λ R , θ] of polynomials depending of ten-dimensional even pure spinors λ L , λ R and a ten-dimensional odd spinor θ. In other words, this means that the components λ L , λ R obey the pure spinor constraint (1) and the components θ α are free Grassmann variables. We define a differential acting on A by the formula:

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Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra

Cited by 24 publications

References 6 publications

The Rational Higher Structure of M‐theory

The Rational Higher Structure of M‐theory

Division algebras and supersymmetry III

Cohomology Ring of the BRST Operator Associated to the Sum of Two Pure Spinors

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